<h2>The ring of Siegel modular forms of degree 2 with respect to 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(4)$</a>
</h2>

<div class="literature">
  <ul>
    <li><span class="name">H. Aoki, T.Ibukiyama: </span>Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Internat. J. Math. 16 (2005), 249-279, <a href="http://www.ams.org/mathscinet-getitem?mr==2130626">MR2130626</a></li>
  </ul>
</div>

<p>
  By the results in <span class="name">
    Aoki and Ibukiyama</span>
  (Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Internat. J. Math. 16 (2005), 249-279, 
  <a href="http://www.ams.org/mathscinet-getitem?mr=MR2130626">MR2130626</a>),
  the ring <script type="math/tex">M_{*}(\Gamma_0(4))</script> of Siegel modular forms of degree 2 with respect to the group 
<a class="knowl-title" knowl="mf.siegel.group.gamma0">$\Gamma_0(4)$</a>
is generated by the following generators, which involve the usage of 
<a class="knowl-title" knowl="mf.siegel.theta_constants">theta constants</a>.

<ul>
<li>
<a href="{{ url_for('not_yet_implemented') }}">$X$</a>, a form of weight 2,
with formula
$$X = ((\theta_{0000})^4+(\theta_{0001})^4+(\theta_{0010})^4+(\theta_{0011})^4)/4.$$
</li>
<li>
<a href="{{ url_for('not_yet_implemented') }}">$X(2\tau)$</a>, a form  of weight 2.
</li>
<li>
<a href="{{ url_for('not_yet_implemented') }}">$f_2(2\tau)$</a>, a form  of weight 4,
with formula
$$f_2 = (\theta_{0000})^4.$$
</li>
<li>
<a href="{{ url_for('not_yet_implemented') }}">$K(2\tau)$</a>, a form  of weight 6,
with formula
$$K = (\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111})^2/4096.$$
</li>
<li>
<a href="{{ url_for('not_yet_implemented') }}">$f_{11}(2\tau)$</a>, a cusp form of weight 11,
with formula
$$f_{11} = f_6\chi_5,$$
where
<ul>
<li>
$\chi_5=\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011}\theta_{0100}\theta_{0110}\theta_{1000}\theta_{1001}\theta_{1100}\theta_{1111}$
</li>
<li>
$f_6 = ((\theta_{0001})^{4}-(\theta_{0010})^{4})
((\theta_{0001})^{4}-(\theta_{0011})^{4})((\theta_{0010})^{4}-(\theta_{0011})^{4})$.
</li>
</ul>
</li>
</ul>

The generators $X, X(2\tau),f_2(2\tau), K(2\tau)$ are algebraically independent.
Denote $B={\Bbb C}[X, X(2\tau),f_2(2\tau), K(2\tau)]$.
Then the ring of modular forms is

$$M(\Gamma_0(4)) =
B + Y(2\tau) B + f_{11}(2\tau)(B + Y(2\tau) B),$$
where
$Y = (\theta_{0000}\theta_{0001}\theta_{0010}\theta_{0011})^2.$



The ideal of cusp forms is ..??...


